A167759 -id:A167759 - cp's OEIS frontend

A168086 Numbers k such that d(k) is not an isolated number.

1, 4, 9, 16, 24, 25, 30, 36, 40, 42, 48, 49, 54, 56, 64, 66, 70, 78, 80, 81, 88, 100, 102, 104, 105, 110, 112, 114, 120, 121, 128, 130, 135, 136, 138, 144, 152, 154, 162, 165, 168, 169, 170, 174, 176, 182, 184, 186, 189, 190, 192, 195, 196, 208, 210, 216, 222

Offset: 1

Juri-Stepan Gerasimov, Nov 18 2009

nonn

A000005 = A167759 U A168086. Where 0,1,3,5,7,8,9,10,11,13,14,15,16,17,19,20,21,22,.. are nonisolated numbers A167707. The nonisolated numbers of divisors of n. The positions of isolated numbers in A000005.

			A000005(a(1)=1)=1, A000005(a(2)=4)=3, A000005(a(3)=9)=3.

Cf. A000005, A167759, A167707.

A000005(a(n)) = nonisolated number.

Corrected (132, 140, 148 removed, 152 inserted etc.) by R. J. Mathar, Jun 04 2010

A275740 Sums of the next n consecutive nonsquare integers.

Original entry on oeis.org

0, 2, 8, 21, 46, 83, 136, 210, 306, 426, 575, 758, 972, 1223, 1519, 1855, 2236, 2669, 3156, 3694, 4290, 4956, 5678, 6467, 7332, 8269, 9278, 10368, 11548, 12804, 14148, 15593, 17126, 18753, 20485, 22325, 24262, 26308, 28481, 30756, 33148

Offset: 0

Olivier Gérard, Aug 07 2016

Row sums of nonsquare integers (A000037), seen as a regular triangle:
.
2   | 2,
8   | 3,  5,
21  | 6,  7,  8,
46  | 10, 11, 12, 13,
83  | 14, 15, 17, 18, 19,
136 | 20, 21, 22, 23, 24, 26,
210 | 27, 28, 29, 30, 31, 32, 33,
306 | 34, 35, 37, 38, 39, 40, 41, 42,
...
The equivalent for all integers are A006003 (starting from 1), A229183 (starting from 2) and A027480 (starting from 0).
There are several sequences close to nonsquares whose sum of groups of n terms starts like this sequence, notably A028761, A158276, A167759.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000037, A006003, A229183, A027480.

R:= 0: s:= 1:
for n from 1 to 100 do
  if floor(sqrt(s+n)) = floor(sqrt(s)) then
    R:= R, n*s + n*(n+1)/2; s:= s+n;
  else
    R:= R, n*s + n*(n+1)/2 - floor(sqrt(s+n))^2 + s+n+1; s:= s+n+1;
  fi
od:
R; # Robert Israel, Oct 02 2022

Table[Sum[
  i + Floor[1/2 + Sqrt[i]], {i, n (n - 1)/2 + 1, (n + 1) (n)/2}], {n,
  0, 40}]
Join[{0},Module[{nn=1000,nsi,len},nsi=Select[Range[nn],!IntegerQ[Sqrt[#]]&];len=Floor[ (Sqrt[ 8*Length[nsi]+1]-1)/2];Total/@TakeList[nsi,Range[len]]]] (* Harvey P. Dale, Jan 04 2024 *)

Definition clarified by Harvey P. Dale, Jan 04 2024

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A168086 Numbers k such that d(k) is not an isolated number.

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A275740 Sums of the next n consecutive nonsquare integers.

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